Principal Lines on an Ellipsoid in a Minkowski Three-Dimensional Space

The description of principal lines of the ellipsoid on the 3-dimensional Minkowski space is established. A global principal parametrization of a triple orthogonal system of quadrics is also achieved, and the focal set of the ellipsoid is sketched.

Key words: principal lines, configuration principal, Minkowski three-dimensional space, ellipsoid, triple orthogonal system

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A Miquel-Steiner Transformation

Each complete quadrilateral has three Miquel-Steiner points. Any triangle together with an arbitrarily chosen point not on a triangle side also defines a complete quadrilateral, and thus, this pivot point defines three Miquel-Steiner points. These three Miquel points form a triangle which is perspective with the base triangle. The mapping that assigns to the pivot point the uniquely defined perspector is a quadratic and not involutive Cremona transformation and shall be called Miquel-Steiner transformation. We shall study the action of the Miquel-Steiner transformation and its inverse.

Key words: Miquel points, quadrilateral, triangle, quadratic Cremona transformation

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On the Geometry of Spherical Trochoids

We provide a synthetic study of the top-views of spherical trochoids. These projections turn out to be higher trochoids, i.e., curves generated by the superposition of more than two rotations. Special shapes of these trochoids show up for special choices of the spherical radii of the rolling circles. A relation to closed algebraic curves of constant width is shown. These curves allow for a kinematic generation.

Key words: spherical trochoid, rolling, evolute, involute, curve of constant width

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Locus Curves in Triangle Families

In this article, we observe a one-parameter triangle family, where two vertices are fixed and the third vertex lies on a given line. For this family of triangles, we observe the loci of centroids, orthocenters, circumcenters, incenters, excenters and some triangle elements associated to these triangle points.

Key words:family of triangles, centroid, orthocenter, circumcenter, incenter, excenter

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Curves in 3-dimensional Minkowski Space

In this paper curves in threedimensional Minkowski space were analyzed and the main differences in local theory of curves in Euclidean and Minkowski space were emphasized. Special attention is paid to curves with no Euclidean counterpart. There are numerous examples of studied curves whose graphic representations were made by Mathematica software.

Key words: Minkowski space, spacelike curve, timelike curve, lightlike curve

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Circles Related to a Complete Quadrangle

This paper presents an overview of some properties of a complete quadrangle ABCD in the Euclidean plane. We study the circles with diameters AB, AC, AD, BC, BD, and CD, as well as the pedal triangles and the pedal circles of the points A, B, C, D with respect to the triangles BCD, ACD, ABD and ABC, respectively. The presented results are known in literature, but here we prove them using a single method.

Key words: complete quadrangle, pedal triangles, pedal circles

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